# Investigate the difference of the product of the diagonally opposite corners of a certain shape, drawn on a 10x10 grid with the individual squares numbered off 1 to 100.

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Introduction

Maths Coursework Aim: I task is to investigate the difference of the product of the diagonally opposite corners of a certain shape, drawn on a 10x10 grid with the individual squares numbered off 1 to 100 I will start off by working this out on a 2x2 square and from there I will begin to investigate varying the length and width of rectangles and other squares. I will aim to investigate the differences for different sized rectangles and squares that are aligned differently on the grid. I will record findings and ideas as I proceed. Method: To make things easier for me I will break up each investigation in to sections, I will start with: 2x2 square I will concentrate on one particular aspect at a time. * What is the difference between the products of the corners? * Is the difference the same for a square or rectangle the same shape anywhere on the grid? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 2 x 11 = 22 1 x 12 = 12 Diff = 10 Are all the differences 10? ...read more.

Middle

From my results I can draw a table: Length (L) Height (H) Area (A) Difference (D) 2 2 4 10 2 3 6 20 2 4 8 30 2 5 10 40 2 6 12 50 What do I notice? The area increases by 2 each time. This is because the length is always being multiplied by the height of 2. The difference increases by 10 each time. This is because the grid is 10x10 and makes it so if you choose any number, the next number vertical down from it is always 10 higher. Possible Formulas L-1 x 10 i.e. The length subtracted by one then multiplied by 10. But this formula must be tested with other rectangles with a length greater than 2 because it is highly probable it won't work due to height not being mentioned in the equation. L-1 x 5H i.e. The length subtracted by one then 5 multiplied by the height. Do these rules really work with other rectangles? I will now look at rectangles where the length is 3 squares across and see if the same patterns apply and investigate whether the rules I came up with are true for rectangles of any size. I believe I shall require only on example of each and will proceed on this principal. 3x4 rectangle 1 2 3 4 11 12 13 14 21 22 23 24 4 x 21 = 84 1 x 24 = 24 Diff = 60 3x5 rectangle 1 2 3 4 5 11 12 13 14 15 21 22 23 24 25 5 x 21 = 105 1 x 25 = 25 Diff = 80 Here the difference increases in multiples of 20. ...read more.

Conclusion

This is correct as from my previous working I know that on a 10x10 grid the difference is 60 (10x6) I can now build a table from my results: L H ?G 2 3 2 2 4 3 3 5 6 From this table I can see that between 2x3 and 2x4 that ? only increases by 1. This must be because the height has been increased also by 1. This tells me that the height must be connected to the difference. I will now work out how: * How can I get 2 from 2&3, and 3 from 2&4 using the same formula? If I subtract 1 from each of 2&3 I get L=1, H=2 1 x 2 = 2 The answers to both these short sums give me the figures to multiply G with to get the difference. I predict that this will work for 2&4: L = 2-1 = 1 H = 4-1 = 3 1 x 3 = 3 Wrote as an equation this method is L-1 x H-1 x G. Which is my previous rule of L-1 x G(H-1) just moved around, which proves that it is correct. Summarising: I have found the rule for any size rectangle or square on any sized grid which is: L-1 x G(H-1) Evaluation I believe I have completed my aims and succeeded in attaining the goals I had set. I have found the rule to work out the difference for a rectangle of any size on any grid. I have done this through progressive investigation and through the use of algebraic methods. I am happy with my outcome as I am convinced it is right. Nick Claydon Page 1 ...read more.

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